Related papers: Finite Difference Approximation with ADI Scheme fo…
We derive the two-dimensional Keller-Segel equation from a stochastic system of $N$ interacting particles in the case of sub-critical chemosensitivity $\chi < 8 \pi$. The Coulomb interaction force is regularised with a cutoff of size $N^{-…
We propose a new high-order alternating direction implicit (ADI) finite difference scheme for the solution of initial-boundary value problems of convection-diffusion type with mixed derivatives and non-constant coefficients, as they arise…
In this paper, we propose and analyze a time-stepping method for the time fractional Allen-Cahn equation. The key property of the proposed method is its unconditional stability for general meshes, including the graded mesh commonly used for…
We develop a second-order accurate central scheme for the two-dimensional hyperbolic system of in-homogeneous conservation laws. The main idea behind the scheme is that we combine the well-balanced deviation method with the Kurganov-Tadmor…
We construct, analyse and assess various schemes of second order of accuracy in space and time for model advection-diffusion-reaction differential equations. The constructed schemes are meant to be of practical use in solving industrial…
Partial differential equations (PDEs) describing thermodynamically isolated systems typically possess conserved quantities (like mass, momentum, and energy) and dissipated quantities (like entropy). Preserving these conservation and…
A quasi-second order scheme is developed to obtain approximate solutions of the shallow water equationswith bathymetry. The scheme is based on a staggered finite volume scheme for the space discretization:the scalar unknowns are located in…
The two-dimensional unsteady coupled Burgers' equations with moderate to severe gradients, are solved numerically using higher-order accurate finite difference schemes; namely the fourth-order accurate compact ADI scheme, and the…
In this article, a nonlinear fractional Cable equation is solved by a two-grid algorithm combined with finite element (FE) method. A temporal second-order fully discrete two-grid FE scheme, in which the spatial direction is approximated by…
In this paper, we propose and study a stochastic aggregation-diffusion equation of the Keller-Segel (KS) type for modeling the chemotaxis in dimensions $d=2,3$. Unlike the classical deterministic KS system, which only allows for…
We study the solutions of the two-dimensional Keller-Segel system describing chemotaxis. The Keller-Segel system as well as the properties of the blow-up set has been extensively studied. In this paper we obtain generalized solutions for…
We consider a two dimensional parabolic-elliptic Keller-Segel equation with a logistic forcing and a fractional diffusion of order $\alpha$. We obtain existence of global in time regular solution for arbitrary initial data with no size…
In this paper a simple, effective adaptation of Alternating Direction Implicit (ADI) time discretization schemes is proposed for the numerical pricing of American-style options under the Heston model via a partial differential…
We introduce stochastic models of chemotaxis generalizing the deterministic Keller-Segel model. These models include fluctuations which are important in systems with small particle numbers or close to a critical point. Following Dean's…
This paper proposes and analyzes an implicit-explicit BDF-Galerkin scheme of second order for the time-dependent nonlinear thermistor problem. For this, we combine the second-order backward differentiation formula with special extrapolation…
This paper proposes some efficient and accurate adaptive two-grid (ATG) finite element algorithms for linear and nonlinear partial differential equations (PDEs). The main idea of these algorithms is to utilize the solutions on the $k$-th…
This paper introduces alternating-direction implicit (ADI) solvers of higher order of time-accuracy (orders two to six) for the compressible Navier-Stokes equations in two- and three-dimensional curvilinear domains. The higher-order…
In this paper we propose an explicit two-level conservative scheme based on a TE/TM like splitting of the field components in time. Its dispersion properties are adjusted to accelerator problems. It is simpler and faster than the implicit…
We propose a structure-preserving finite difference scheme for the Cahn-Hilliard equation with a dynamic boundary condition using the discrete variational derivative method (DVDM). In this approach, it is important and essential how to…
We investigate the numerical discretization of a two-stream kinetic system with an internal state, such system has been introduced to model the motion of cells by chemotaxis. This internal state models the intracellular methylation level.…